過去の記録

2011年06月24日(金)

15:00-16:30 数理科学研究科棟(駒場) 128号室関口次郎 氏 (東京農工大)
A Schwarz map of Appell's $F_2$ whose monodromy group isrelated to the reflection group of type $D_4$ (JAPANESE)

[ 講演概要 ]
The system of differential equations for Appell's hypergeometric function $F_2(a,b,b',c,c';x,y)$ has four fundamental solutions. Let $u_1,u_2,u_3,u_4$ be such solutions. If the monodromy group of the system is finite, the closure of the image of the Schwarz map $U(x,y)=(u_1(x,y),u_2(x,y),u_3(x,y),u_4(x,y))$ is a hypersurface $S$ of the 3-dimensional projective space ${\\bf P}^3$. Then $S$ is defined by $P(u_1,u_2,u_3,u_4)=0$ for a polynomial $P(t_1,t_2,t_3,t_4)$. It is M. Kato (Univ. Ryukyus) who determined the parameter $a,b,b',c,c'$ such that the monodromy group of the system for $F_2(a,b,b',c,c';x,y)$ is finite. It follows from his result that such a group is the semidirect product of an irreducible finite reflection group $G$ of rank four by an abelian group. In this talk, we treat the system for $F_2(a,b,b',c,c';x,y)$ with $(a,b,b',c,c')=(1/2,1/6,-1/6,1/3,2/3$. In this case, the monodromy group is the semidirect group of $G$ by $Z/3Z$, where $G$ is the reflection group of type $D_4$. The polynomial $P(t_1,t_2,t_3,t_4)$ in this case is of degree four. There are 16 ordinary singular points in the hypersurface $S$. In the rest of my talk, I explain the background of the study.

2011年06月15日(水)

[ 講演概要 ]
I would like to talk about my recent work jointly with A. Marmora on a product formula for $p$-adic epsilon factors. In 80's Deligne conjectured that a constant appearing in the functional equation of $L$-function of $\\ell$-adic lisse sheaf can be written by means of local contributions, and proved some particular cases. This conjecture was proven later by Laumon, and was used in the Lafforgue's proof of the Langlands' program for functional filed case. In my talk, I would like to prove a $p$-adic analog of this product formula.

(本講演は「東京パリ数論幾何セミナー」として、インターネットによる東大数理とIHESとの双方向同時中継で行います。)

2011年06月14日(火)

[ 講演概要 ]
For polarized algebraic manifolds, the concept of K-stability introduced by Tian and Donaldson is conjecturally strongly correlated to the existence of constant scalar curvature metrics (or more generally extremal K\\"ahler metrics) in the polarization class. This is known as Donaldson-Tian-Yau's conjecture. Recently, a remarkable progress has been made by many authors toward its solution. In this talk, I'll discuss the topic mainly with emphasis on the existence part of the conjecture.

[ 講演概要 ]
We consider Schr\\"odinger operators defined on star graphs with Kirchhoff boundary conditions. Under suitable decay conditions on the potential, we construct a complete set of eigenfunctions to obtain spectral representations of the operator. The results are applied to give a time dependent formulation of the scattering theory. Also we use the spectral representation to determine an integral equation of Marchenko which is fundamental to enter into the inverse scattering problems.

2011年06月07日(火)

[ 講演概要 ]
(joint with Christopher Hacon) In this talk, we will address the problem on given a log canonical variety, how we compactify it. Our approach is via MMP. The result has a few applications. Especially I will explain the one on the moduli of stable schemes.If time permits, I will also talk about how a similar approach can be applied to give a proof of the existence of log canonical flips and a conjecture due to Kollár on the geometry of log centers.

[ 講演概要 ]
It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

[ 講演概要 ]
It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

[ 講演概要 ]
For an arbitrary variety we define a multiplier ideal by using Mather discrepancy.This ideal coincides with the usual multiplier ideal if the variety is normal and complete intersection.In the talk I will show a local vanishing theorem for this ideal and as corollaries we obtain restriction theorem, subadditivity theorem, Skoda type theorem, and Briancon-Skoda type theorem.

2011年06月02日(木)

[ 講演概要 ]
In the representation theory of quantum groups at roots of unity, it isoften assumed that the parameter $q$ is a primitive $n$-th root of unitywhere $n$ is a odd prime number. However, there has recently beenincreasing interest in the the cases where $n$ is an even integer ---for example, in the study of logarithmic conformal field theories, or inknot invariants. In this talk,we work out a fairly detailed study on the category of finite dimensionalmodules of the restricted quantum $¥overline{U}_q(¥mathfrak{sl}_2)$ where$q$ is a $2p$-th root of unity, $p¥ge2$.

2011年05月31日(火)

[ 講演概要 ]
This is a joint work with Yusuke Tokunaga. Let $M$ be an $N$ dimensional compact connected smooth Riemannian manifold without boundary and let $\\mathcal{E}^{r}(M,M)$ be the space of $C^{r}$ expandig maps endowed with $C^{r}$ topology. We show thateach of the following properties for element $T$ in $\\mathcal{E} ^{1}(M,M)$ is generic.\\begin{itemize}\\item[(1)] $T$ has a unique measure with maximum total exponent.\\item[(2)] Any measure with maximum total exponent for $T$ has zero entropy.\\item[(3)] Any measure with maximum total exponent for $T$ is fully supported.\\end{itemize}On the contrary, we show that for $r\\ge 2$, a generic element in $\\mathcal{E}^{r}(M,M)$ has no fully supported measures with maximum total exponent.

16:30-17:30 数理科学研究科棟(駒場) 126号室栗原 大武 氏 (東北大学大学院理学研究科)
On character tables of association schemes based on attenuatedspaces (JAPANESE)

[ 講演概要 ]
An association scheme is a pair of a finite set $X$and a set of relations $\\{R_i\\}_{0\\le i\\le d}$on $X$ which satisfies several axioms of regularity.The notion of association schemes is viewed as some axiomatizedproperties of transitive permutation groups in terms of combinatorics, and also the notion of association schemes is regarded as a generalization of the subring of the group ring spanned by the conjugacy classes of finite groups.Thus, the theory of association schemes had been developed in thestudy of finite permutation groups and representation theory.To determine the character tables of association schemes is animportant first step to a systematic study of association schemes, and is helpful toward the classification of those schemes.

In this talk, we determine the character tables of association schemes based on attenuated spaces.These association schemes are obtained from subspaces of a givendimension in attenuated spaces.

2011年05月30日(月)

[ 講演概要 ]
$f:X\\to Y$ be an algebraic fiber space with generic geometric fiber $F$, $\\dim X=n$ and $\\dim Y=m$. Then Iitaka's $C_{n,m}$ conjecture states $$\\kappa (X)\\geq \\kappa (Y)+\\kappa (F).$$ In particular, if $X$ is a variety with $\\kappa(X)=0$ and $f: X \\to Y$ is the Albanese map, then Ueno conjecture that $\\kappa(F)=0$. One can regard Ueno’s conjecture an important test case of Iitaka’s conjecture in general.

These conjectures are of fundamental importance in the classification of higher dimensional complex projective varieties. In a recent joint work with Hacon, we are able to prove Ueno’s conjecture and $C_{n,m}$ conjecture holds when $Y$ is of maximal Albanese dimension. In this talk, we will introduce some relative results and briefly sketch the proof.